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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="abcmiz_1.html#V3" title="ABCMIZ_1:attr.3">initialized</a>  <a href="abcmiz_1.html#NM5" title="ABCMIZ_1:NM.5">ConstructorSignature</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">e</font> being  <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font> holds <br/> ( <font color="Olive" title="b1">e</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a>  iff  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">e</font> be   <a href="abcmiz_1.html#NM6" title="ABCMIZ_1:NM.6">expression</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">e</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a>  iff  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<span class="kw">thus </span><a NAME="E1:190"/>
( <font color="Maroon" title="c2">e</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a>  implies  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c2">e</font> is  <a href="abcmiz_1.html#V4" title="ABCMIZ_1:attr.4">ground</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="190_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:190_1"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="abcmiz_1.html#D26" target="_self" title="ABCMIZ_1:def.26">ABCMIZ_1:def 26</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<span class="kw">hence </span><a NAME="E2:190_1"/>
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T1" target="_self" title="ABCMIZ_1:th.1">Th1</a>, <a class="ref" href="xboole_1.html#T3" title="XBOOLE_1:th.3">XBOOLE_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:190"/><span class="lab"><font color="Green" title="E94">A1</font></span>: 
 <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E3:190"/><span class="lab"><font color="Green" title="E95">A2</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="abcmiz_1.html#D26" target="_self" title="ABCMIZ_1:def.26">ABCMIZ_1:def 26</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span>;<br/>
<a NAME="E5:190"/><span class="lab"><font color="Green" title="E96">A3</font></span>: 
 ex <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:190"><span class="lab"><font color="Green" title="E95">A2</font></span></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></span>;<br/>
<span class="lab"><font color="Green" title="E97">A4</font></span>:  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
<br/>.= 
<span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="abcmiz_1.html#K6" title="ABCMIZ_1:func.6">a_Type</a> ,<a href="abcmiz_1.html#K7" title="ABCMIZ_1:func.7">an_Adj</a> ,<a href="abcmiz_1.html#K8" title="ABCMIZ_1:func.8">a_Term</a> </span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#D9" target="_self" title="ABCMIZ_1:def.9">Def9</a></span>
;<br/>
<a NAME="E7:190"/><span class="lab"><font color="Green" title="E98">A5</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#T28" title="MSAFREE3:th.28">MSAFREE3:28</a></span>;<br/>
<a NAME="E8:190"/><span class="lab"><font color="Green" title="E99">A6</font></span>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="abcmiz_1.html#K7" title="ABCMIZ_1:func.7">an_Adj</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#D25" target="_self" title="ABCMIZ_1:def.25">Def25</a></span>;<br/>
<a NAME="E9:190"/><span class="lab"><font color="Green" title="E100">A7</font></span>: 
<span class="p1">(<span class="default"><a href="abcmiz_1.html#K28" title="ABCMIZ_1:func.28">MSVars</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="abcmiz_1.html#K6" title="ABCMIZ_1:func.6">a_Type</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#D25" target="_self" title="ABCMIZ_1:def.25">Def25</a></span>;<br/>
<a NAME="E10:190"/><span class="lab"><font color="Green" title="E101">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K13" title="ABCMIZ_1:func.13">an_Adj</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:190"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E8:190"><span class="lab"><font color="Green" title="E99">A6</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></span>;<br/>
<a NAME="E11:190"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">C</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c2">e</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:190"><span class="lab"><font color="Green" title="E98">A5</font></span></a>, <a class="txt" href="#E9:190"><span class="lab"><font color="Green" title="E100">A7</font></span></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:190"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:190"><span class="lab"><font color="Green" title="E94">A1</font></span></a>, <a class="txt" href="#E5:190"><span class="lab"><font color="Green" title="E96">A3</font></span></a>, <a class="txt" href="#E6:190"><span class="lab"><font color="Green" title="E97">A4</font></span></a>, <a class="txt" href="#E10:190"><span class="lab"><font color="Green" title="E101">A8</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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